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Theorem distgp 24123
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
distgp.1 𝐵 = (Base‘𝐺)
distgp.2 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
distgp ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Proof of Theorem distgp
StepHypRef Expression
1 simpl 482 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp)
2 simpr 484 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵)
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
43fvexi 6921 . . . . 5 𝐵 ∈ V
5 distopon 23020 . . . . 5 (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵))
64, 5ax-mp 5 . . . 4 𝒫 𝐵 ∈ (TopOn‘𝐵)
72, 6eqeltrdi 2847 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵))
8 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
93, 8istps 22956 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
107, 9sylibr 234 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp)
11 eqid 2735 . . . . . 6 (-g𝐺) = (-g𝐺)
123, 11grpsubf 19050 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1312adantr 480 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
144, 4xpex 7772 . . . . 5 (𝐵 × 𝐵) ∈ V
154, 14elmap 8910 . . . 4 ((-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1613, 15sylibr 234 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)))
172, 2oveq12d 7449 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵))
18 txdis 23656 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵))
194, 4, 18mp2an 692 . . . . . 6 (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)
2017, 19eqtrdi 2791 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵))
2120oveq1d 7446 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽))
22 cndis 23315 . . . . 5 (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2314, 7, 22sylancr 587 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2421, 23eqtrd 2775 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2516, 24eleqtrrd 2842 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
268, 11istgp2 24115 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
271, 10, 25, 26syl3anbrc 1342 1 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  𝒫 cpw 4605   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Basecbs 17245  TopOpenctopn 17468  Grpcgrp 18964  -gcsg 18966  TopOnctopon 22932  TopSpctps 22954   Cn ccn 23248   ×t ctx 23584  TopGrpctgp 24095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-topgen 17490  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-tx 23586  df-tmd 24096  df-tgp 24097
This theorem is referenced by: (None)
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